
Koornwinder Class \(V\)  hypergroups:
In this case \(\mathcal {K}:=\lbrace (n,k)\in {N}_0^2: n\ge k\rbrace \)  and
the characters are given by
\(P_{\bf {n}}(x,y):=P_{(n,k)}^{\alpha ,\beta ,\gamma ,\eta }(x,y):=P_{n-k}^{(\alpha ,\beta )}(x) P_{k}^{(\gamma ,\eta )}(y),\hspace{14.22636pt}{\bf {n}}=(n,k), \) 
where \(P_{n}^{(\alpha ,\beta )}\)  denote the Jacobi polynomials, \((\alpha ,\beta )\) ,
\((\gamma , \eta )\in V\) , \(P_{\bf {n}}^{\alpha ,\beta ,\gamma ,\eta }(1,1)=1\) ,
and
\(V:=\lbrace (\alpha ,\beta )\in {R}^2:&\;\alpha \ge \beta >-1, (\alpha +\beta +1)(\alpha +\beta +4)^2(\alpha +\beta +6)\\ &\ge (\alpha -\beta )^2\cdot (\alpha ^2-2\alpha \beta +\beta ^2-5\alpha -5\beta -30)\rbrace .\) 

The support of the Plancherel measure
\(d\pi (x,y)=(1-x)^\alpha (1+x)^\beta (1-y)^\gamma (1+y)^\eta dx dy\)  is
\(D:=\lbrace (x,y)| -1\le x\le 1, -1\le y\le 1\rbrace .\) 
Since \( | P_n^{(\alpha ,\beta )}(y)|=\mathcal {O}(n^{-\alpha -\frac{1}{2}})\)  as \(n\rightarrow \infty \)  [1]},
we have
\(|P_{(n,n)}^{\alpha ,\beta ,\gamma ,\eta }(x,y)|= |P_{n}^{(\gamma ,\eta )}(y)|\rightarrow 0\hspace{28.45274pt}(n\rightarrow \infty )\) 
when \((x,y)\in [-1,1]\times (-1,1)\)  and \(\alpha \) , \(\eta >-\frac{1}{2}\) . So, from
Theorem REF  it follows that \(\mathcal {K}\)  is not \(\alpha _{(x,y)}\) -amenable.
For \((x,y)\in \lbrace (-1, 1), (1, -1)(-1, -1)\rbrace \) , if \(\alpha >\beta \)  and
\(\gamma >\eta \) , \(\alpha =\beta \)  and \(\gamma >\eta \) , or \(\alpha >\beta \)  and \(\gamma =\eta \) 
since
\(P_n^{(\alpha ,\beta )}(-1)=(-1)^n\left(\begin{array}{c}n+\beta \\n \\\end{array}\right){\big {/}}\left(\begin{array}{c}n+\alpha \\n \\\end{array}\right),\) 
we have \(|P_{(2n,n)}^{\alpha ,\beta ,\gamma ,\eta }(x,y)|\rightarrow 0\) 
as
\(n\rightarrow \infty ,\)  hence \(\mathcal {K}\)  is not \(\alpha _{(x,y)}\) -amenable.
The hypergroup \(\mathcal {K}\)  is, in fact, the product of two Jacobi polynomial
hypergroups with parameters \((\alpha ,\beta )\)  and \((\gamma ,\eta )\)  on \({N}_0\) 
[2]}.
Theorem REF  combined with [3]} implies
that \(\ell ^1({N}_0)\)  is amenable if
and only if \(\alpha =\beta =\gamma =\eta =-\frac{1}{2}\) .
Thus, since \(\ell ^1(\mathcal {K})\cong \ell ^1({N}_0)\otimes _p\ell ^1({N}_0)\) ,
the algebra \(\ell ^1(\mathcal {K})\)  is amenable and its maximal ideals have
b.a.i.; see [4]}, [5]}.
Consequently, Theorem REF  results in the
\(\alpha _{(x,y)}\) -amenability of
\(\mathcal {K}\)  for \((x,y)\in D\)  and \(x,y= \pm 1\) .
Remark 2.1.2 

Let \((x, y_0)\in [-1,1]\times [-1, 1]\) 
be as above fixed. For \( \gamma >\frac{1}{2}\)  one
can show that the usual derivation of the Fourier
transform gives a rise to a nonzero bounded \(\alpha _{(x,y_0)}\) -derivation
on \(\ell ^1(\mathcal {K})\) . So, it follows from
Remark REF  that \(\mathcal {K}\)  is not
\(\alpha _{(x,y_0)}\) -amenable and \(\lbrace \alpha _{(x,y_0)}\rbrace \)  is not a spectral set.

Similar to the previous case, one can show that hypergroups
of Koornwinder class III, VI, and some
related hypergroups in two variables which are mentioned in
[6]} are not \(\alpha _{\bf {x}}\) -amenable
if \(\alpha _{\bf {x}}\ne 1\) .



Disc Polynomial Hypergroups:
For \(\alpha ^{\prime }\ge 0\)  the disc polynomials
\(P_{m,n}^{\alpha ^{\prime }} (z, \bar{z})=\left\lbrace \begin{array}{ll}P_n^{(\alpha ^{\prime }, m-n) }(2z\bar{z}-1)z^{m-n}, & \hbox{for $m\ge n$,} \\P_m^{(\alpha ^{\prime }, n-m) }(2z\bar{z}-1)z^{n-m}, & \hbox{for $n\ge m$,}\end{array}\right.\) 
induce a hypergroup structure on \(\mathcal {K}:={N}_0^2\) . The support
of the Plancherel measure
with the density
\((z_1, z_2)\rightarrow c_{\alpha ^{\prime }} (1-|z_1|^2)^{\alpha ^{\prime }} \)  is
\(\mathcal {D}:=\lbrace  (z_1, z_2)\in {C}^2:\; z_2=\bar{z}_1, \; |z_1|<1\rbrace \) .
From Theorem REF  and
\(P_{n,n}^{\alpha ^{\prime }}(z, \bar{z})&= P_n^{(\alpha ^{\prime }, 0) }(2z\bar{z}-1)= P_n^{(\alpha ^{\prime }, 0) }(2|z|^2-1)\\&=\mathcal {O}(n^{-\alpha ^{\prime }-1/2})\hspace{28.45274pt}(z\in \mathcal {D}),\) 

as \(n\rightarrow \infty \) , we infer that \(\mathcal {K}\)  is \(\alpha _z\) -amenable if
and only
if \(\alpha _z=1\) .
Observe that \(\mathcal {H}:= \lbrace (n, n):\; n\in {N}_0\rbrace \)  is a supernormal subhypergroup of
\(\mathcal {K}\)  which is
isomorphic to the Jacobi hypergroup
with the character set
\(\lbrace P_{n,n}^{\alpha ^{\prime }}(x) \rbrace _{n\in {N}_0}\) . In
this case we see also
that \(\mathcal {H}\)  is \(\alpha _x\) -amenable
if and only if \(\alpha _x=1\) 
despite the fact that for
every \(x\in (-1,1)\)  the singleton \(\lbrace \alpha _x\rbrace \)  is a
spectral for \(\mathcal {H}\) 
if
\(\alpha ^{\prime } <\frac{1}{2}\) ; see [3]}.
In other words, if \(\alpha ^{\prime } <\frac{1}{2}\) 
then every
bounded \(\alpha _x\) -derivation on \(\ell ^1(\mathcal {H})\) 
is zero, however \(\mathcal {H}\) 
is only 1-amenable.

